Properties

Label 6001.2.a.c
Level $6001$
Weight $2$
Character orbit 6001.a
Self dual yes
Analytic conductor $47.918$
Analytic rank $0$
Dimension $121$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6001,2,Mod(1,6001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6001, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6001.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6001 = 17 \cdot 353 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6001.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(47.9182262530\)
Analytic rank: \(0\)
Dimension: \(121\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 121 q + 9 q^{2} + 13 q^{3} + 127 q^{4} + 21 q^{5} + 19 q^{6} - 13 q^{7} + 24 q^{8} + 134 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 121 q + 9 q^{2} + 13 q^{3} + 127 q^{4} + 21 q^{5} + 19 q^{6} - 13 q^{7} + 24 q^{8} + 134 q^{9} - q^{10} + 40 q^{11} + 41 q^{12} + 14 q^{13} + 32 q^{14} + 49 q^{15} + 135 q^{16} - 121 q^{17} + 28 q^{18} + 34 q^{19} + 64 q^{20} + 34 q^{21} - 18 q^{22} + 37 q^{23} + 54 q^{24} + 128 q^{25} + 91 q^{26} + 55 q^{27} - 28 q^{28} + 45 q^{29} + 30 q^{30} + 67 q^{31} + 47 q^{32} + 40 q^{33} - 9 q^{34} + 59 q^{35} + 138 q^{36} - 16 q^{37} + 30 q^{38} + 37 q^{39} + 14 q^{40} + 89 q^{41} + 33 q^{42} + 16 q^{43} + 90 q^{44} + 83 q^{45} - 9 q^{46} + 135 q^{47} + 96 q^{48} + 128 q^{49} + 71 q^{50} - 13 q^{51} + 47 q^{52} + 52 q^{53} + 90 q^{54} + 93 q^{55} + 69 q^{56} - 4 q^{57} + 5 q^{58} + 170 q^{59} + 78 q^{60} - 2 q^{61} + 46 q^{62} - 10 q^{63} + 182 q^{64} + 50 q^{65} + 68 q^{66} + 46 q^{67} - 127 q^{68} + 97 q^{69} + 46 q^{70} + 191 q^{71} + 57 q^{72} - 12 q^{73} + 68 q^{74} + 86 q^{75} + 108 q^{76} + 62 q^{77} - 10 q^{78} + 130 q^{80} + 149 q^{81} + 14 q^{82} + 83 q^{83} + 126 q^{84} - 21 q^{85} + 132 q^{86} + 50 q^{87} - 42 q^{88} + 144 q^{89} + 9 q^{90} + 13 q^{91} + 50 q^{92} + 43 q^{93} + 41 q^{94} + 82 q^{95} + 110 q^{96} - 3 q^{97} + 36 q^{98} + 89 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.80099 1.23587 5.84553 −1.08657 −3.46165 −3.96716 −10.7713 −1.47263 3.04347
1.2 −2.75847 −2.11283 5.60915 3.70679 5.82818 −2.88565 −9.95572 1.46406 −10.2251
1.3 −2.69106 2.90048 5.24179 1.92081 −7.80536 −1.20204 −8.72383 5.41280 −5.16901
1.4 −2.63511 −2.54509 4.94380 −1.75130 6.70660 −0.500367 −7.75725 3.47750 4.61488
1.5 −2.61678 −3.27387 4.84754 0.197154 8.56700 −1.64878 −7.45139 7.71823 −0.515910
1.6 −2.60140 −0.594038 4.76726 2.91382 1.54533 0.0965456 −7.19873 −2.64712 −7.57999
1.7 −2.58940 0.995654 4.70501 0.868437 −2.57815 −0.888525 −7.00437 −2.00867 −2.24873
1.8 −2.54984 3.38084 4.50169 −1.07763 −8.62059 1.13594 −6.37892 8.43005 2.74778
1.9 −2.53200 0.341990 4.41103 2.95840 −0.865919 4.74675 −6.10474 −2.88304 −7.49068
1.10 −2.48167 1.61831 4.15867 −3.04222 −4.01610 0.428830 −5.35710 −0.381079 7.54977
1.11 −2.44940 0.192595 3.99956 1.46027 −0.471742 −3.96634 −4.89773 −2.96291 −3.57679
1.12 −2.29935 −2.65633 3.28702 −0.0208883 6.10783 4.05842 −2.95931 4.05607 0.0480296
1.13 −2.25146 1.98627 3.06909 4.11708 −4.47201 2.76168 −2.40701 0.945268 −9.26946
1.14 −2.24876 −0.983386 3.05693 −2.62112 2.21140 0.682123 −2.37679 −2.03295 5.89428
1.15 −2.22966 1.54651 2.97139 −1.67307 −3.44820 −3.79558 −2.16587 −0.608295 3.73037
1.16 −2.18200 −1.07317 2.76112 −3.39433 2.34166 −4.74202 −1.66076 −1.84830 7.40643
1.17 −2.13704 −0.352862 2.56694 0.777890 0.754080 0.156355 −1.21157 −2.87549 −1.66238
1.18 −2.08608 1.06648 2.35172 −3.22907 −2.22475 0.560788 −0.733710 −1.86263 6.73610
1.19 −2.05839 1.11600 2.23697 1.51993 −2.29716 0.698036 −0.487784 −1.75454 −3.12862
1.20 −2.05373 −1.71490 2.21782 −3.91336 3.52195 1.58894 −0.447336 −0.0591109 8.03700
See next 80 embeddings (of 121 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.121
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(17\) \(1\)
\(353\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6001.2.a.c 121
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6001.2.a.c 121 1.a even 1 1 trivial